I have never seen a cogent case made for the interest of inconsistent
systems for foundations of mathematics. Yet I still think that there is
much more to be said against them, and something new and interesting will
likely emerge in arguing against them.
I am new to this topic, so I will start with exploratory stuff, and expect
experts to further the discussion. At some point, if new results emerge,
then I will move them to my numbered postings.
Perhaps the most historically important inconsistent system developed for
f.o.m. is the full comprehension axiom scheme, which I will take in its
form with epsilon only (no equality), absurdity, and the connectives and,
or, if then, iff.
With the usual axioms and rules of classical logic, not only do we get a
contradiction, but we also derive all formulas. This is also true if we use
the axioms and rules of intuitionistic logic.
However, we can weaken the logic to avoid the rule that from absurdity we
can derive everything. Then it is not clear that CA = full comprehension
axiom scheme derives all formulas.
What axioms and rules of classical logic are appropriate here to use with
CA so that we don't derive all formulas?
To address this question, it seems best to first remove absurdity entirely
from CA and understand what is going on. CA without any absurdity will be
denoted by CA(no). We will use the usual classical introduction and
elimination rules for and, or, if then, and construe iff as an
abbreviation.
Let alpha be the sentence (forall x,y)(x in y).
We argue in CA(no). Let A = {x: x in x implies alpha).
LEMMA 1. CA(no) proves: A in A implies alpha.
LEMMA 2. CA(no) proves: A in A.
LEMMA 3. CA(no) proves: alpha.
LEMMA 4. CA(no) proves every formula in the language of CA(no).
I leave it to the experts whether this is convincing, whether this is new,
and what implications it has for various foundational and philosophical
enterprises? And what are the next things to look at?
Harvey Friedman
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